The new Amp with its HDMI-ARC connection (or optical, with the
optical to HDMI adapter - not included with the Amp) will definitely have the response time you're looking for, for your projector audio. It's very likely that there will be a noticeable delay using the analog Line In on the Connect:Amp. If that's a key thing for you, I'd say to get the Amp and sell the Connect:Amp.
I will note that I don't have a Connect:Amp, but I've experienced such delays using the Line In on a Connect.
What did you decide? Just a thought but can’t you line up the audio by going into TV settings and adjusting the audio delay? I could be wrong but I think most TVs these days have that option.